There are two ways to use "to like", with different translations:
to like s.th. = etwas mögen
to like to do s.th. = etwas gern(e) tun
Example: Ich mag Fussball. Ich spiele gerne Fussball. ("Ich mag Fussball spielen" is acceptable in colloquial speech).
OTOH, "möchten" is used with a verb: "Ich möchte Fussball spielen". (Again, "Ich möchte ein Eis" is used in colloquial speech, esp. by children.).
So "gerne" is an adverb that indicates that the action is agreeable. If you add "gern" to "mögen" or "möchten", it works in the same way: it emphasizes that one really likes to do it, just like "besonders". Therefore
"Was magst du?", "Was magst du gern?", "Was magst du besonders gern?"
"Was möchtest du tun?", "Was möchtest du gern tun?", "Was möchtest du besonders gern tun?"
all mean the same thing, with different degrees of emphasis. Same for the answer.